Abstract

The rapid and accurate computation of the Jacobian matrix, which is usually computationally intensive, is of critical importance for the reconstruction problem of fluorescent molecular tomography (FMT). An extension of the Green’s function method for the Jacobian matrix computation suitable for two coupled differential equations is proposed, in combination with the parallel forward computing strategy for FMT image reconstruction. For further acceleration of the reconstruction process without significant quality reduction of the results, we also propose to reconstruct the FMT image on an adaptively refined mesh generated with a priori information incorporated. Experimental results demonstrate that the speed of the reconstruction process can be significantly improved with the proposed overall algorithm.

© 2007 Optical Society of America

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  1. A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
    [CrossRef]
  2. R. B. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental Fluorescence Tomography of Tissues With Noncontact Measurements," IEEE Trans. Med. Imaging 23, 492-500 (2004).
    [CrossRef] [PubMed]
  3. A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, "Fluorescence optical diffusion tomography," Appl. Opt. 42, 3081-3094 (2003).
    [CrossRef] [PubMed]
  4. A. D. Klose, V. Ntziachristos, and A. H. Hielscher, "The inverse source problem based on the radiative transfer equation in optical molecular imaging," J. Comput. Phys. 202, 323-345 (2005).
    [CrossRef]
  5. F. Fedele, M. J. Eppstein, J. P. Laible, A. Godavarty, and E. M. Sevick-Muraca, "Fluorescence photon migration by the boundary element method," J. Comput. Phys. 210, 1-24 (2005).
    [CrossRef]
  6. A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
    [CrossRef] [PubMed]
  7. A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, Part 1:hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
    [CrossRef] [PubMed]
  8. S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
    [CrossRef]
  9. Q. Lin, Basic Text Book of Numerical Solution Method for Differential Equations, 2nd ed. (Science Press, 2003).
  10. J. Li, "Research on positive definiteness of matrix," Practice Cogn. Math. 2, 59-63 (1995).
  11. D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, "Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media," Appl. Opt. 36, 2260-2272 (1997).
    [CrossRef]
  12. S. R. Arridge and M. Schweiger, "A general framework for iterative reconstruction algorithms in optical tomography using a finite element method," in Computational Radiology and Imaging: Therapy and Diagnosis, IMA Volumes in Mathematics and Its Applications (Springer-Verlag, 1998).
  13. Z. Pan, Text Book of Maths and Physics Methods (Nankai University Press, 1993).
  14. S. R. Arridge and J. C. Hebden, "Optical imaging in medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
    [CrossRef] [PubMed]
  15. A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, "Adaptive finite element based tomography for fluorescence optical imaging in tissue," Opt. Express 12, 5402-5417 (2004).
    [CrossRef] [PubMed]
  16. X. Pang, "The research on positive definite complex matrix," J. Shandong Univ. 38, 66-69 (2003).

2005

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, "The inverse source problem based on the radiative transfer equation in optical molecular imaging," J. Comput. Phys. 202, 323-345 (2005).
[CrossRef]

F. Fedele, M. J. Eppstein, J. P. Laible, A. Godavarty, and E. M. Sevick-Muraca, "Fluorescence photon migration by the boundary element method," J. Comput. Phys. 210, 1-24 (2005).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

2004

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, Part 1:hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

R. B. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental Fluorescence Tomography of Tissues With Noncontact Measurements," IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, "Adaptive finite element based tomography for fluorescence optical imaging in tissue," Opt. Express 12, 5402-5417 (2004).
[CrossRef] [PubMed]

2003

X. Pang, "The research on positive definite complex matrix," J. Shandong Univ. 38, 66-69 (2003).

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, "Fluorescence optical diffusion tomography," Appl. Opt. 42, 3081-3094 (2003).
[CrossRef] [PubMed]

Q. Lin, Basic Text Book of Numerical Solution Method for Differential Equations, 2nd ed. (Science Press, 2003).

1999

S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

1998

S. R. Arridge and M. Schweiger, "A general framework for iterative reconstruction algorithms in optical tomography using a finite element method," in Computational Radiology and Imaging: Therapy and Diagnosis, IMA Volumes in Mathematics and Its Applications (Springer-Verlag, 1998).

1997

1995

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
[CrossRef]

J. Li, "Research on positive definiteness of matrix," Practice Cogn. Math. 2, 59-63 (1995).

1993

Z. Pan, Text Book of Maths and Physics Methods (Nankai University Press, 1993).

Abdoulaev, G. S.

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, Part 1:hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Arridge, S. R.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

S. R. Arridge and M. Schweiger, "A general framework for iterative reconstruction algorithms in optical tomography using a finite element method," in Computational Radiology and Imaging: Therapy and Diagnosis, IMA Volumes in Mathematics and Its Applications (Springer-Verlag, 1998).

S. R. Arridge and J. C. Hebden, "Optical imaging in medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

Bangerth, W.

Bluestone, A. Y.

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, Part 1:hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Boas, D. A.

Bouman, C. A.

Chance, B.

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
[CrossRef]

Chen, A. U.

Eppstein, M. J.

F. Fedele, M. J. Eppstein, J. P. Laible, A. Godavarty, and E. M. Sevick-Muraca, "Fluorescence photon migration by the boundary element method," J. Comput. Phys. 210, 1-24 (2005).
[CrossRef]

Fedele, F.

F. Fedele, M. J. Eppstein, J. P. Laible, A. Godavarty, and E. M. Sevick-Muraca, "Fluorescence photon migration by the boundary element method," J. Comput. Phys. 210, 1-24 (2005).
[CrossRef]

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Godavarty, A.

F. Fedele, M. J. Eppstein, J. P. Laible, A. Godavarty, and E. M. Sevick-Muraca, "Fluorescence photon migration by the boundary element method," J. Comput. Phys. 210, 1-24 (2005).
[CrossRef]

Hebden, J. C.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Hebden, "Optical imaging in medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

Hielscher, A. H.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, "The inverse source problem based on the radiative transfer equation in optical molecular imaging," J. Comput. Phys. 202, 323-345 (2005).
[CrossRef]

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, Part 1:hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Joshi, A.

Klose, A. D.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, "The inverse source problem based on the radiative transfer equation in optical molecular imaging," J. Comput. Phys. 202, 323-345 (2005).
[CrossRef]

Laible, J. P.

F. Fedele, M. J. Eppstein, J. P. Laible, A. Godavarty, and E. M. Sevick-Muraca, "Fluorescence photon migration by the boundary element method," J. Comput. Phys. 210, 1-24 (2005).
[CrossRef]

Lasker, J.

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, Part 1:hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Li, J.

J. Li, "Research on positive definiteness of matrix," Practice Cogn. Math. 2, 59-63 (1995).

Lin, Q.

Q. Lin, Basic Text Book of Numerical Solution Method for Differential Equations, 2nd ed. (Science Press, 2003).

Millane, R. P.

Milstein, A. B.

Ntziachristos, V.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, "The inverse source problem based on the radiative transfer equation in optical molecular imaging," J. Comput. Phys. 202, 323-345 (2005).
[CrossRef]

R. B. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental Fluorescence Tomography of Tissues With Noncontact Measurements," IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

Oh, S.

Paithankar, D. Y.

Pan, Z.

Z. Pan, Text Book of Maths and Physics Methods (Nankai University Press, 1993).

Pang, X.

X. Pang, "The research on positive definite complex matrix," J. Shandong Univ. 38, 66-69 (2003).

Patterson, M. S.

Pogue, B. W.

Ripoll, J.

R. B. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental Fluorescence Tomography of Tissues With Noncontact Measurements," IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

Schulz, R. B.

R. B. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental Fluorescence Tomography of Tissues With Noncontact Measurements," IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

Schweiger, M.

S. R. Arridge and M. Schweiger, "A general framework for iterative reconstruction algorithms in optical tomography using a finite element method," in Computational Radiology and Imaging: Therapy and Diagnosis, IMA Volumes in Mathematics and Its Applications (Springer-Verlag, 1998).

Sevick-Muraca, E. M.

Stewart, M.

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, Part 1:hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Webb, K. J.

Yodh, A.

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
[CrossRef]

Zhang, Q.

Appl. Opt.

IEEE Trans. Med. Imaging

R. B. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental Fluorescence Tomography of Tissues With Noncontact Measurements," IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

Inverse Probl.

S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

J. Biomed. Opt.

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, Part 1:hypercapnia," J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

J. Comput. Phys.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, "The inverse source problem based on the radiative transfer equation in optical molecular imaging," J. Comput. Phys. 202, 323-345 (2005).
[CrossRef]

F. Fedele, M. J. Eppstein, J. P. Laible, A. Godavarty, and E. M. Sevick-Muraca, "Fluorescence photon migration by the boundary element method," J. Comput. Phys. 210, 1-24 (2005).
[CrossRef]

Opt. Express

Phys. Med. Biol.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Hebden, "Optical imaging in medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

Phys. Today

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (1995).
[CrossRef]

Other

Q. Lin, Basic Text Book of Numerical Solution Method for Differential Equations, 2nd ed. (Science Press, 2003).

J. Li, "Research on positive definiteness of matrix," Practice Cogn. Math. 2, 59-63 (1995).

S. R. Arridge and M. Schweiger, "A general framework for iterative reconstruction algorithms in optical tomography using a finite element method," in Computational Radiology and Imaging: Therapy and Diagnosis, IMA Volumes in Mathematics and Its Applications (Springer-Verlag, 1998).

Z. Pan, Text Book of Maths and Physics Methods (Nankai University Press, 1993).

X. Pang, "The research on positive definite complex matrix," J. Shandong Univ. 38, 66-69 (2003).

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Figures (7)

Fig. 1
Fig. 1

Schematic illustration of the principle of FMT.

Fig. 2
Fig. 2

Model of reconstruction.

Fig. 3
Fig. 3

Model of a priori image.

Fig. 4
Fig. 4

(a) Adaptively refined mesh, (b) globally coarse mesh, (c) globally fine mesh.

Fig. 5
Fig. 5

Reconstructed spatial map of absorption coefficient due to fluorophore μ a x f on the adaptively refined mesh with (a) our parallel algorithm and (b) sequential method.

Fig. 6
Fig. 6

Reconstructed spatial map of absorption coefficient due to fluorophore μ a x f on the adaptively refined mesh with (a) Green’s function method and (b) traditional perturbation method.

Fig. 7
Fig. 7

Reconstructed spatial map of absorption coefficient due to fluorophore μ a x f with Green’s function method on (a) adaptively refined mesh, (b) globally coarse mesh, (c) globally fine mesh.

Tables (4)

Tables Icon

Table 1 Optical and Fluorescent Properties

Tables Icon

Table 2 Comparison of Performance of Algorithms

Tables Icon

Table 3 Comparison of Performance of Methods

Tables Icon

Table 4 Comparison of Reconstruction for Different Meshes

Equations (56)

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( D x + k x ) Φ x = S x ,
( D m + k m ) Φ m = β Φ x .
D x = 1 3 ( μ a x i + μ a x f + μ s x ) ,
D m = 1 3 ( μ a m i + μ a m f + μ s m ) ,
k x = i ω c + μ a x i + μ a x f ,
k m = i ω c + μ a m i + μ a m f ,
β = η μ a x f 1 i ω τ ,
n [ D x Φ x ( r ) ] + b x Φ x ( r ) = 0 for all r Ω ,
n [ D m Φ m ( r ) ] + b m Φ m ( r ) = 0 for all r Ω ,
( D m + k m ) Φ = 1 .
( D m + k m ) Φ β Φ x = β Φ x .
Φ m = f ( Φ , Φ x ) .
v h = k = 1 N c k φ k .
a Ω h ( u h , v h ) x , m = ( f x , m , v h ) Ω h ,
a Ω h ( u h , v h ) x , m = Ω h [ D x , m ( u h v h ) + k x , m u h v h ] d Ω + Γ h b x , m u h v h d s ,
( f x , m , v h ) Ω h = Ω h f x , m v h d Ω ,
f x = S x , f m = β Φ x ;
A x , m Φ x , m = S x , m ,
S x , m = [ ( f x , m , φ 1 ) Ω h ( f x , m , φ N ) Ω h ] ,
A x , m = [ a Ω h ( φ 1 , φ 1 ) x , m a Ω h ( φ N , φ 1 ) x , m a Ω h ( φ 1 , φ N ) x , m a Ω h ( φ N , φ N ) x , m ] .
a Ω h ( φ i , φ j ) x , m = Ω h D x , m φ i φ j d Ω + Ω h k x , m φ i φ j d Ω + Γ h b x , m φ i φ j d s .
( D x + K x + B x ) Φ x = S x ,
( D m + K m + B m ) Φ m = S m ,
A x , m = ( D x , m + K x , m + B x , m ) ,
D i j = Ω h D x , m φ i φ j d Ω ,
K i j = Ω h k x , m φ i φ j d Ω ,
B i j = Γ h b x , m φ i φ j d s .
A m H = I ,
Φ m = H S m .
Δ x = J T ( J J T + λ I ) 1 Δ y ,
K x K x + Δ K x , D x D x + Δ D x , Φ x Φ x + Δ Φ x .
A x Δ Φ x = ( Δ D x + Δ K x ) Φ x .
( Δ Φ x ) n = ( G n * ) T ( Δ D x + Δ K x ) Φ x ,
A x * G n * = Q n * ,
J μ a x f , Φ s , n , p = ( Δ Φ x ) n ( Δ μ a x f ) p = ( G n * ) T V x p Φ x ,
( V x p ) i j = Ω k φ p φ i φ j d Ω .
Φ m Φ m + Δ Φ m , S m S m + Δ S m .
Δ Φ m = H Δ Q m ,
( Δ Q m ) i = Ω h η 1 i ω τ Δ μ a x f Φ x φ i d Ω + Ω h η 1 i ω τ μ a x f Δ Φ x φ i d Ω = ( Δ μ a x f ) p Ω h η 1 i ω τ φ p Φ x φ i d Ω + Ω h η 1 i ω τ μ a x f Δ Φ x φ i d Ω .
J μ a x f , Φ m , p = Δ Φ m ( Δ μ a x f ) p = H Δ Q m ( Δ μ a x f ) p .
[ Δ Q m ( Δ μ a x f ) p ] i = ( Δ Q m ) i ( Δ μ a x f ) p = Ω h η 1 i ω τ φ p Φ x φ i d Ω + Ω h η 1 i ω τ μ a x f Δ Φ x ( Δ μ a x f ) p φ i d Ω = Ω h η 1 i ω τ φ p Φ x φ i d Ω + Ω h η 1 i ω τ μ a x f n = 0 N ( Δ Φ x ) n ( Δ μ a x f ) p φ n φ i d Ω .
D ( X ) = E { [ X E ( X ) ] 2 } ,
ε i = ( x i x 0 i ) x 0 i × 100 % , i = 1 , , M ,
MSE = 1 M i = 1 M ( x 0 i x i ) 2 .
a Ω h ( φ i , φ j ) x , m = a Ω h ( φ j , φ i ) x , m .
A x , m T = A x , m .
f 1 , Ω h = { Ω h [ f 2 + ( f x ) 2 + ( f y ) 2 ] d Ω + Γ h f 2 d s } 1 2 .
Re ( V H A x , m V ) = Re [ i , j = 1 N a Ω h ( φ i , φ j ) x , m c ¯ i c j ] = Re [ a Ω h ( i = 1 N c ¯ i φ i , j = 1 N c j φ j ) x , m ] = Re [ a Ω h ( v ¯ h , v h ) x , m ] ,
Re ( a Ω h ( v ¯ h , v h ) x , m ) = Re ( Ω h { D x , m [ v h x 2 + v h y 2 ] + k x , m v h 2 } d Ω + Γ h b x , m v h 2 d s ) γ 1 { Ω h [ v h x 2 + v h y 2 ] d Ω + Γ h v h 2 d s } = 1 2 γ 1 { Ω h [ v h x 2 + v h y 2 ] d Ω + Γ h v h 2 d s } + 1 2 γ 1 { Ω h [ v h x 2 + v h y 2 ] d Ω + Γ h v h 2 d s } ,
v h x 2 = ( a x ) 2 + ( b x ) 2 = 1 4 ( a 2 + b 2 ) [ 4 a 2 ( a x ) 2 + 4 b 2 ( b x ) 2 + 4 a 2 ( b x ) 2 + 4 b 2 ( a x ) 2 ] 1 4 ( a 2 + b 2 ) [ 4 a 2 ( a x ) 2 + 4 b 2 ( b x ) 2 + 8 a b b x a x ] 1 4 ( a 2 + b 2 ) [ 4 a 2 ( a x ) 2 + 4 b 2 ( b x ) 2 + 8 a b b x a x ] ,
( v h x ) 2 = ( a 2 + b 2 x ) 2 = 1 4 ( a 2 + b 2 ) [ ( a 2 ) x + ( b 2 ) x ] 2 = 1 4 ( a 2 + b 2 ) [ 2 a a x + 2 b b x ] 2 = 1 4 ( a 2 + b 2 ) [ 4 a 2 ( a x ) 2 + 4 b 2 ( b x ) 2 + 8 a b b x a x ] .
v h x 2 ( v h x ) 2 .
v h y 2 ( v h y ) 2 .
Ω [ ( v x ) 2 + ( v y ) 2 ] d Ω + Γ v 2 d s c Ω v 2 d Ω ,
Re [ a Ω h ( v ¯ h , v h ) x , m ] 1 2 γ 1 { Ω h [ ( v h x ) 2 + ( v h y ) 2 ] d Ω + Γ h v h 2 d s } + 1 2 γ 1 { Ω h [ v h x 2 + v h y 2 ] d Ω + Γ h v h 2 d s } 1 2 γ 1 c Ω h v h 2 d Ω + 1 2 γ 1 { Ω h [ v h x 2 + v h y 2 ] d Ω + Γ h v h 2 d s } = 1 2 γ 1 { c Ω h v h 2 d Ω + Ω h [ v h x 2 + v h y 2 ] d Ω + Γ h v h 2 d s } 1 2 γ 1 { c Ω h v h 2 d Ω + Ω h [ ( v h x ) 2 + ( v h y ) 2 ] d Ω + Γ h v h 2 d s } γ v h 1 , Ω h 2 ,
Re ( V H A x , m V ) = Re [ a Ω h ( v ¯ h , v h ) x , m ] γ v h 1 , Ω h 2 > 0 , V 0 .

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